Editing Double planet (section)

=== Tug-of-war value ===

[[Isaac Asimov]] suggested a distinction between planet–moon and double-planet structures based in part on what he called a "[[tug of war (astronomy)|tug-of-war]]" value, which does not consider their relative sizes.<ref name="Asimov">[[Isaac Asimov|Asimov, Isaac]] (1975). "Just Mooning Around", collected in ''[http://www.univeros.com/usenet/cache/alt.binaries.ebooks/10.000.SciFi.and.Fantasy.Ebooks/Isaac%20Asimov/Isaac%20Asimov%20-%20Of%20Time%20and%20Space%20and%20Other%20Things.pdf Of Time and Space, and Other Things] {{Webarchive|url=https://web.archive.org/web/20180107175033/http://www.univeros.com/usenet/cache/alt.binaries.ebooks/10.000.SciFi.and.Fantasy.Ebooks/Isaac%20Asimov/Isaac%20Asimov%20-%20Of%20Time%20and%20Space%20and%20Other%20Things.pdf |date=2018-01-07 }}''. Avon. Formula derived on p. 89 of book. p. 55 of .pdf file. Retrieved 2012-01-20.</ref> This quantity is simply the ratio of the force exerted on the smaller body by the larger (primary) body to the force exerted on the smaller body by the Sun. This can be shown to equal

<math display=block>\text{tug-of-war value} = \frac{m_\mathrm{p}}{m_\mathrm{s}} \cdot \left( \frac{d_\mathrm{s}}{d_\mathrm{p}} \right)^2</math>

where {{math|''m''<sub>p</sub>}} is the mass of the primary (the larger body), {{math|''m''<sub>s</sub>}} is the mass of the Sun, {{math|''d''<sub>s</sub>}} is the distance between the smaller body and the Sun, and {{math|''d''<sub>p</sub>}} is the distance between the smaller body and the primary.<ref name="Asimov"/> The tug-of-war value does not rely on the mass of the satellite (the smaller body).

This formula actually reflects the relation of the [[gravitation]]al effects on the smaller body from the larger body and from the Sun. The tug-of-war figure for Saturn's moon [[Titan (moon)|Titan]] is 380, which means that Saturn's hold on Titan is 380 times as strong as the Sun's hold on Titan. Titan's tug-of-war value may be compared with that of Saturn's moon [[Phoebe (moon)|Phoebe]], which has a tug-of-war value of just 3.5; that is, Saturn's hold on Phoebe is only 3.5 times as strong as the Sun's hold on Phoebe.

Asimov calculated tug-of-war values for several satellites of the planets. He showed that even the largest gas giant, Jupiter, had only a slightly better hold than the Sun on its outer captured satellites, some with tug-of-war values not much higher than one. In nearly every one of Asimov's calculations the tug-of-war value was found to be greater than one, so in those cases the Sun loses the tug-of-war with the planets. The one exception was Earth's Moon, where the Sun wins the tug-of-war with a value of 0.46, which means that Earth's hold on the Moon is less than half as strong as the Sun's. Asimov included this with his other arguments that Earth and the Moon should be considered a binary planet.<ref name="Asimov"/>

{{Blockquote|We might look upon the Moon, then, as neither a true satellite of the Earth nor a captured one, but as a planet in its own right, moving about the Sun in careful step with the Earth. From within the Earth–Moon system, the simplest way of picturing the situation is to have the Moon revolve about the Earth; but if you were to draw a picture of the orbits of the Earth and Moon about the Sun exactly to scale, you would see that the Moon's orbit is everywhere concave toward the Sun. It is always "falling toward" the Sun. All the other satellites, without exception, "fall away" from the Sun through part of their orbits, caught as they are by the superior pull of their primary planets{{spaced ndash}}but not the Moon.<ref name="Asimov"/><ref name="Aslaksen">{{Cite web |title=The Orbit of the Moon around the Sun is Convex! |last=Aslaksen |first=Helmer |url=http://www.math.nus.edu.sg/aslaksen/teaching/convex.html |year=2010 |location=National University of Singapore |publisher=Department of Mathematics |access-date=2012-01-23 |archive-url=https://web.archive.org/web/20130116204505/http://www.math.nus.edu.sg/aslaksen/teaching/convex.html |archive-date=2013-01-16 |url-status=dead }}</ref><ref name="PoV" group="Footnote">Asimov uses the term "[[Wikt:concave|concave]]" to describe the Earth–Moon orbital pattern around the Sun, whereas Aslaksen uses "[[Wikt:convex|convex]]" to describe the exact same pattern. Which term one uses relies solely upon the perspective of the observer. From the point-of-view of the Sun, the Moon's orbit is concave; from outside the Moon's orbit, say, from planet Mars, it is convex.</ref>| Isaac Asimov}}
See the [[Orbit of the Moon#Path of Earth and Moon around Sun|Path of Earth and Moon around Sun]] section in the "Orbit of the Moon" article for a more detailed explanation.

This definition of double planet depends on the pair's distance from the Sun. If the Earth–Moon system happened to orbit farther away from the Sun than it does now, then Earth would win the tug of war. For example, at the orbit of Mars, the Moon's tug-of-war value would be 1.05. Also, several tiny moons discovered since Asimov's proposal would qualify as double planets by this argument. Neptune's small outer moons [[Neso (moon)|Neso]] and [[Psamathe (moon)|Psamathe]], for example, have tug-of-war values of 0.42 and 0.44, less than that of Earth's Moon. Yet their masses are tiny compared to Neptune's, with an estimated ratio of 1.5{{e|-9}} ({{frac|700,000,000}}) and 0.4{{e|-9}} ({{frac|2,500,000,000}}).